# How do you write a sum in expanded form?

Mar 17, 2015

Perhaps you just mean to convert it from "summation form" ("sigma form") to a written out form?

For something like $\setminus {\sum}_{i = 1}^{n} {i}^{2}$, the summation symbol $\setminus \Sigma$ just means to "add up". Putting an $i = 1$ underneath the summation symbol means to start the value of $i$ at 1. It is then assumed that $i$ keeps increasing by 1 until it reaches $i = n$, where $n$ is the number above the summation symbol. The ${i}^{2}$ represents the formula for the terms that get added, first when $i = 1$, then $i = 2$, then $i = 3$, etc..., until $i = n$.

Therefore, the answer would be $\setminus {\sum}_{i = 1}^{n} {i}^{2} = {1}^{2} + {2}^{2} + {3}^{2} + \setminus \cdots + {\left(n - 1\right)}^{2} + {n}^{2}$.

This example is interesting in that there is a shortcut formula for adding up the first $n$ squares: it equals

$\setminus \frac{n \left(n + 1\right) \left(2 n + 1\right)}{6} = \setminus \frac{1}{3} {n}^{3} + \setminus \frac{1}{2} {n}^{2} + \setminus \frac{1}{6} n .$

You should take the time to check that this works when, for instance, $n = 5$.